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Separable differential equations are a special type of differential equation that can be manipulated to separate the function of one variable from the function of another variable. The goal is to achieve a format where each side of the equation involves only one variable. This means we aim for an arrangement like:

• \( f(y) \, dy = g(x) \, dx \)
• Both sides can then be integrated separately, enabling us to find the solution.

To solve a separable differential equation such as \( \sec x \, dy + x \cos^2 y \, dx = 0 \), we first move the terms around so that all occurrences of \( y \) are on one side of the equation and all occurrences of \( x \) are on the other. By doing so, we where each side has terms dependent only on one variable, allowing us to integrate each part separately.
This process is what was undertaken in the original exercise to separate \( dy \) and \( dx \), resulting in equation: \[ \frac{dy}{\cos^2 y} = -x \frac{dx}{\sec x} \]. The goal moving forward is to evaluate these integrals once the variables are separated.

Integration by parts is a technique used to integrate products of functions. It is especially handy when one of the functions is easier to differentiate, while the other is easier to integrate. This technique is derived from the product rule of differentiation and is expressed as:
\[ \int u \, dv = uv - \int v \, du \]When applied correctly, this method can simplify complex integrals into more manageable expressions. For instance, in solving the equation \( \int -x \cos x \, dx \), integration by parts is used. Here:

• Choose \( u = x \) because it can be straightforwardly differentiated.
• Set \( dv = \cos x \, dx \), making it easy to integrate \( v = \sin x \).

With these choices, we derive the integral: \[- \int x \cos x \, dx = -\left( x \sin x - \int \sin x \, dx \right) = -x \sin x + \cos x + C \]This integral is much simpler than the original expression we started with, and combining it with other integrated terms potentially offers a complete solution to the original differential equation.

Trigonometric integrals contain functions of trigonometric expressions such as sine, cosine, or tangent, and often require specific techniques or identities for solutions.
When integrating trigonometric functions, using identities can greatly simplify the process. For example:

• The identity \( \cos^2 y = \frac{1 + \cos 2y}{2} \) is often used.
• This approach transforms a complex integral into a more simple form by enabling partial fraction decompositions or direct integration techniques.

In the solution to the given differential equation, simplifying \( \int \cos^2 y \, dy \) with this identity allows us to integrate it as:
\[ \int \frac{1 + \cos 2y}{2} \, dy = \frac{1}{2} \left( y + \frac{\sin 2y}{2} \right) + C \]This changes the problem from a potentially complicated trig integral into the manageable task of integrating simpler functions, enabling the combination of both the left and right side solutions easily after integration.